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Broad Topics > Using, Applying and Reasoning about Mathematics > Working systematically

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Problem Solving, Using and Applying and Functional Mathematics

Age 5 to 18 Challenge Level:

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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Gr8 Coach

Age 11 to 14 Challenge Level:

Can you coach your rowing eight to win?

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Pole Star Sudoku 2

Age 11 to 16 Challenge Level:

This Sudoku, based on differences. Using the one clue number can you find the solution?

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The Naked Pair in Sudoku

Age 7 to 16

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

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Quadruple Sudoku

Age 11 to 16 Challenge Level:

Four small numbers give the clue to the contents of the four surrounding cells.

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One Out One Under

Age 14 to 16 Challenge Level:

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

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9 Weights

Age 11 to 14 Challenge Level:

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

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LOGO Challenge - Triangles-squares-stars

Age 11 to 16 Challenge Level:

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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Difference Sudoku

Age 14 to 16 Challenge Level:

Use the differences to find the solution to this Sudoku.

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Olympic Logic

Age 11 to 16 Challenge Level:

Can you use your powers of logic and deduction to work out the missing information in these sporty situations?

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More on Mazes

Age 7 to 14

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

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Instant Insanity

Age 11 to 18 Challenge Level:

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

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Twin Line-swapping Sudoku

Age 14 to 16 Challenge Level:

A pair of Sudoku puzzles that together lead to a complete solution.

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Magic Caterpillars

Age 14 to 18 Challenge Level:

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

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Ratio Sudoku 2

Age 11 to 16 Challenge Level:

A Sudoku with clues as ratios.

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Intersection Sudoku 1

Age 11 to 16 Challenge Level:

A Sudoku with a twist.

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Twin Corresponding Sudoku

Age 11 to 18 Challenge Level:

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

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Crossing the Bridge

Age 11 to 14 Challenge Level:

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

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An Introduction to Magic Squares

Age 7 to 16

Find out about Magic Squares in this article written for students. Why are they magic?!

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Integrated Sums Sudoku

Age 11 to 16 Challenge Level:

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

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Twin Corresponding Sudokus II

Age 11 to 16 Challenge Level:

Two sudokus in one. Challenge yourself to make the necessary connections.

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Spot the Card

Age 14 to 16 Challenge Level:

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

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Twin Chute-swapping Sudoku

Age 14 to 18 Challenge Level:

A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?

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Magnetic Personality

Age 7 to 16 Challenge Level:

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

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LOGO Challenge - the Logic of LOGO

Age 11 to 16 Challenge Level:

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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Addition Equation Sudoku

Age 11 to 14 Challenge Level:

You need to find the values of the stars before you can apply normal Sudoku rules.

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Tea Cups

Age 7 to 14 Challenge Level:

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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Integrated Product Sudoku

Age 11 to 16 Challenge Level:

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

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Wallpaper Sudoku

Age 11 to 16 Challenge Level:

A Sudoku that uses transformations as supporting clues.

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Warmsnug Double Glazing

Age 11 to 14 Challenge Level:

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

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Latin Squares

Age 11 to 18

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

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Seasonal Twin Sudokus

Age 11 to 16 Challenge Level:

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

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Cinema Problem

Age 11 to 14 Challenge Level:

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

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LOGO Challenge - Sequences and Pentagrams

Age 11 to 18 Challenge Level:

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

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Tetrahedra Tester

Age 11 to 14 Challenge Level:

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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When Will You Pay Me? Say the Bells of Old Bailey

Age 11 to 14 Challenge Level:

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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Difference Dynamics

Age 14 to 18 Challenge Level:

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

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Sticky Numbers

Age 11 to 14 Challenge Level:

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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Rainstorm Sudoku

Age 14 to 16 Challenge Level:

Use the clues about the shaded areas to help solve this sudoku

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Rectangle Outline Sudoku

Age 11 to 16 Challenge Level:

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

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Bent Out of Shape

Age 14 to 18 Challenge Level:

An introduction to bond angle geometry.

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Inky Cube

Age 7 to 14 Challenge Level:

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

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First Connect Three for Two

Age 7 to 14 Challenge Level:

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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Building with Longer Rods

Age 7 to 14 Challenge Level:

A challenging activity focusing on finding all possible ways of stacking rods.

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I've Submitted a Solution - What Next?

Age 5 to 18

In this article, the NRICH team describe the process of selecting solutions for publication on the site.

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Twin Corresponding Sudoku III

Age 11 to 16 Challenge Level:

Two sudokus in one. Challenge yourself to make the necessary connections.

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Multiples Sudoku

Age 11 to 14 Challenge Level:

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

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Star Product Sudoku

Age 11 to 16 Challenge Level:

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

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Colour Islands Sudoku

Age 11 to 14 Challenge Level:

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

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Quadruple Clue Sudoku

Age 11 to 16 Challenge Level:

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.